Properties

Label 3330.2.a.bg.1.3
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.91729\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.91729 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.91729 q^{7} -1.00000 q^{8} -1.00000 q^{10} -6.51056 q^{11} -0.813457 q^{13} -2.91729 q^{14} +1.00000 q^{16} +2.51056 q^{17} +0.406728 q^{19} +1.00000 q^{20} +6.51056 q^{22} -5.02112 q^{23} +1.00000 q^{25} +0.813457 q^{26} +2.91729 q^{28} +5.32401 q^{29} -8.75186 q^{31} -1.00000 q^{32} -2.51056 q^{34} +2.91729 q^{35} -1.00000 q^{37} -0.406728 q^{38} -1.00000 q^{40} -6.34513 q^{41} -7.32401 q^{43} -6.51056 q^{44} +5.02112 q^{46} +5.42784 q^{47} +1.51056 q^{49} -1.00000 q^{50} -0.813457 q^{52} +2.34513 q^{53} -6.51056 q^{55} -2.91729 q^{56} -5.32401 q^{58} +1.42784 q^{59} -1.32401 q^{61} +8.75186 q^{62} +1.00000 q^{64} -0.813457 q^{65} -5.42784 q^{67} +2.51056 q^{68} -2.91729 q^{70} -14.6480 q^{71} -11.0211 q^{73} +1.00000 q^{74} +0.406728 q^{76} -18.9932 q^{77} -1.75870 q^{79} +1.00000 q^{80} +6.34513 q^{82} +7.05476 q^{83} +2.51056 q^{85} +7.32401 q^{86} +6.51056 q^{88} -6.00000 q^{89} -2.37309 q^{91} -5.02112 q^{92} -5.42784 q^{94} +0.406728 q^{95} +2.34513 q^{97} -1.51056 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34} - q^{35} - 3 q^{37} - 3 q^{40} + 9 q^{41} - 11 q^{43} - 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} - 3 q^{50} - 21 q^{53} - 11 q^{55} + q^{56} - 5 q^{58} - 14 q^{59} + 7 q^{61} - 3 q^{62} + 3 q^{64} + 2 q^{67} - q^{68} + q^{70} - 22 q^{71} - 16 q^{73} + 3 q^{74} - 7 q^{77} - 26 q^{79} + 3 q^{80} - 9 q^{82} - 2 q^{83} - q^{85} + 11 q^{86} + 11 q^{88} - 18 q^{89} - 12 q^{91} + 2 q^{92} + 2 q^{94} - 21 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.91729 1.10263 0.551315 0.834297i \(-0.314126\pi\)
0.551315 + 0.834297i \(0.314126\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −6.51056 −1.96301 −0.981503 0.191444i \(-0.938683\pi\)
−0.981503 + 0.191444i \(0.938683\pi\)
\(12\) 0 0
\(13\) −0.813457 −0.225612 −0.112806 0.993617i \(-0.535984\pi\)
−0.112806 + 0.993617i \(0.535984\pi\)
\(14\) −2.91729 −0.779677
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.51056 0.608900 0.304450 0.952528i \(-0.401527\pi\)
0.304450 + 0.952528i \(0.401527\pi\)
\(18\) 0 0
\(19\) 0.406728 0.0933099 0.0466550 0.998911i \(-0.485144\pi\)
0.0466550 + 0.998911i \(0.485144\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 6.51056 1.38806
\(23\) −5.02112 −1.04697 −0.523487 0.852033i \(-0.675369\pi\)
−0.523487 + 0.852033i \(0.675369\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.813457 0.159532
\(27\) 0 0
\(28\) 2.91729 0.551315
\(29\) 5.32401 0.988645 0.494322 0.869279i \(-0.335416\pi\)
0.494322 + 0.869279i \(0.335416\pi\)
\(30\) 0 0
\(31\) −8.75186 −1.57188 −0.785940 0.618303i \(-0.787821\pi\)
−0.785940 + 0.618303i \(0.787821\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.51056 −0.430557
\(35\) 2.91729 0.493111
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −0.406728 −0.0659801
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −6.34513 −0.990943 −0.495471 0.868624i \(-0.665005\pi\)
−0.495471 + 0.868624i \(0.665005\pi\)
\(42\) 0 0
\(43\) −7.32401 −1.11690 −0.558451 0.829538i \(-0.688604\pi\)
−0.558451 + 0.829538i \(0.688604\pi\)
\(44\) −6.51056 −0.981503
\(45\) 0 0
\(46\) 5.02112 0.740323
\(47\) 5.42784 0.791732 0.395866 0.918308i \(-0.370444\pi\)
0.395866 + 0.918308i \(0.370444\pi\)
\(48\) 0 0
\(49\) 1.51056 0.215794
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −0.813457 −0.112806
\(53\) 2.34513 0.322128 0.161064 0.986944i \(-0.448507\pi\)
0.161064 + 0.986944i \(0.448507\pi\)
\(54\) 0 0
\(55\) −6.51056 −0.877883
\(56\) −2.91729 −0.389839
\(57\) 0 0
\(58\) −5.32401 −0.699077
\(59\) 1.42784 0.185889 0.0929447 0.995671i \(-0.470372\pi\)
0.0929447 + 0.995671i \(0.470372\pi\)
\(60\) 0 0
\(61\) −1.32401 −0.169523 −0.0847613 0.996401i \(-0.527013\pi\)
−0.0847613 + 0.996401i \(0.527013\pi\)
\(62\) 8.75186 1.11149
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.813457 −0.100897
\(66\) 0 0
\(67\) −5.42784 −0.663117 −0.331558 0.943435i \(-0.607574\pi\)
−0.331558 + 0.943435i \(0.607574\pi\)
\(68\) 2.51056 0.304450
\(69\) 0 0
\(70\) −2.91729 −0.348682
\(71\) −14.6480 −1.73840 −0.869201 0.494460i \(-0.835366\pi\)
−0.869201 + 0.494460i \(0.835366\pi\)
\(72\) 0 0
\(73\) −11.0211 −1.28992 −0.644962 0.764215i \(-0.723127\pi\)
−0.644962 + 0.764215i \(0.723127\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 0.406728 0.0466550
\(77\) −18.9932 −2.16447
\(78\) 0 0
\(79\) −1.75870 −0.197869 −0.0989346 0.995094i \(-0.531543\pi\)
−0.0989346 + 0.995094i \(0.531543\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.34513 0.700702
\(83\) 7.05476 0.774360 0.387180 0.922004i \(-0.373449\pi\)
0.387180 + 0.922004i \(0.373449\pi\)
\(84\) 0 0
\(85\) 2.51056 0.272308
\(86\) 7.32401 0.789769
\(87\) 0 0
\(88\) 6.51056 0.694028
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.37309 −0.248767
\(92\) −5.02112 −0.523487
\(93\) 0 0
\(94\) −5.42784 −0.559839
\(95\) 0.406728 0.0417295
\(96\) 0 0
\(97\) 2.34513 0.238112 0.119056 0.992888i \(-0.462013\pi\)
0.119056 + 0.992888i \(0.462013\pi\)
\(98\) −1.51056 −0.152589
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.20766 0.816693 0.408346 0.912827i \(-0.366106\pi\)
0.408346 + 0.912827i \(0.366106\pi\)
\(102\) 0 0
\(103\) −8.81346 −0.868416 −0.434208 0.900813i \(-0.642972\pi\)
−0.434208 + 0.900813i \(0.642972\pi\)
\(104\) 0.813457 0.0797660
\(105\) 0 0
\(106\) −2.34513 −0.227779
\(107\) −15.2624 −1.47547 −0.737737 0.675089i \(-0.764105\pi\)
−0.737737 + 0.675089i \(0.764105\pi\)
\(108\) 0 0
\(109\) 4.30290 0.412143 0.206072 0.978537i \(-0.433932\pi\)
0.206072 + 0.978537i \(0.433932\pi\)
\(110\) 6.51056 0.620757
\(111\) 0 0
\(112\) 2.91729 0.275658
\(113\) 9.15859 0.861567 0.430784 0.902455i \(-0.358237\pi\)
0.430784 + 0.902455i \(0.358237\pi\)
\(114\) 0 0
\(115\) −5.02112 −0.468221
\(116\) 5.32401 0.494322
\(117\) 0 0
\(118\) −1.42784 −0.131444
\(119\) 7.32401 0.671391
\(120\) 0 0
\(121\) 31.3874 2.85340
\(122\) 1.32401 0.119871
\(123\) 0 0
\(124\) −8.75186 −0.785940
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.61439 −0.764403 −0.382202 0.924079i \(-0.624834\pi\)
−0.382202 + 0.924079i \(0.624834\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0.813457 0.0713449
\(131\) −13.4278 −1.17320 −0.586598 0.809878i \(-0.699533\pi\)
−0.586598 + 0.809878i \(0.699533\pi\)
\(132\) 0 0
\(133\) 1.18654 0.102886
\(134\) 5.42784 0.468894
\(135\) 0 0
\(136\) −2.51056 −0.215279
\(137\) 14.6903 1.25507 0.627537 0.778587i \(-0.284063\pi\)
0.627537 + 0.778587i \(0.284063\pi\)
\(138\) 0 0
\(139\) 17.3662 1.47299 0.736493 0.676445i \(-0.236481\pi\)
0.736493 + 0.676445i \(0.236481\pi\)
\(140\) 2.91729 0.246556
\(141\) 0 0
\(142\) 14.6480 1.22924
\(143\) 5.29606 0.442879
\(144\) 0 0
\(145\) 5.32401 0.442135
\(146\) 11.0211 0.912114
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 0.207658 0.0170120 0.00850601 0.999964i \(-0.497292\pi\)
0.00850601 + 0.999964i \(0.497292\pi\)
\(150\) 0 0
\(151\) 0.813457 0.0661982 0.0330991 0.999452i \(-0.489462\pi\)
0.0330991 + 0.999452i \(0.489462\pi\)
\(152\) −0.406728 −0.0329900
\(153\) 0 0
\(154\) 18.9932 1.53051
\(155\) −8.75186 −0.702966
\(156\) 0 0
\(157\) 5.32401 0.424903 0.212451 0.977172i \(-0.431855\pi\)
0.212451 + 0.977172i \(0.431855\pi\)
\(158\) 1.75870 0.139915
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −14.6480 −1.15443
\(162\) 0 0
\(163\) 15.2008 1.19062 0.595310 0.803496i \(-0.297029\pi\)
0.595310 + 0.803496i \(0.297029\pi\)
\(164\) −6.34513 −0.495471
\(165\) 0 0
\(166\) −7.05476 −0.547555
\(167\) 12.4826 0.965933 0.482966 0.875639i \(-0.339559\pi\)
0.482966 + 0.875639i \(0.339559\pi\)
\(168\) 0 0
\(169\) −12.3383 −0.949099
\(170\) −2.51056 −0.192551
\(171\) 0 0
\(172\) −7.32401 −0.558451
\(173\) −23.0354 −1.75135 −0.875674 0.482903i \(-0.839583\pi\)
−0.875674 + 0.482903i \(0.839583\pi\)
\(174\) 0 0
\(175\) 2.91729 0.220526
\(176\) −6.51056 −0.490752
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −24.2835 −1.81504 −0.907518 0.420013i \(-0.862026\pi\)
−0.907518 + 0.420013i \(0.862026\pi\)
\(180\) 0 0
\(181\) 2.16543 0.160955 0.0804775 0.996756i \(-0.474355\pi\)
0.0804775 + 0.996756i \(0.474355\pi\)
\(182\) 2.37309 0.175905
\(183\) 0 0
\(184\) 5.02112 0.370162
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −16.3451 −1.19527
\(188\) 5.42784 0.395866
\(189\) 0 0
\(190\) −0.406728 −0.0295072
\(191\) 19.3326 1.39886 0.699429 0.714702i \(-0.253438\pi\)
0.699429 + 0.714702i \(0.253438\pi\)
\(192\) 0 0
\(193\) −20.0422 −1.44267 −0.721336 0.692586i \(-0.756471\pi\)
−0.721336 + 0.692586i \(0.756471\pi\)
\(194\) −2.34513 −0.168370
\(195\) 0 0
\(196\) 1.51056 0.107897
\(197\) −15.6269 −1.11337 −0.556686 0.830723i \(-0.687927\pi\)
−0.556686 + 0.830723i \(0.687927\pi\)
\(198\) 0 0
\(199\) 20.2835 1.43786 0.718931 0.695082i \(-0.244632\pi\)
0.718931 + 0.695082i \(0.244632\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −8.20766 −0.577489
\(203\) 15.5317 1.09011
\(204\) 0 0
\(205\) −6.34513 −0.443163
\(206\) 8.81346 0.614063
\(207\) 0 0
\(208\) −0.813457 −0.0564031
\(209\) −2.64803 −0.183168
\(210\) 0 0
\(211\) 18.7182 1.28862 0.644308 0.764766i \(-0.277146\pi\)
0.644308 + 0.764766i \(0.277146\pi\)
\(212\) 2.34513 0.161064
\(213\) 0 0
\(214\) 15.2624 1.04332
\(215\) −7.32401 −0.499494
\(216\) 0 0
\(217\) −25.5317 −1.73320
\(218\) −4.30290 −0.291429
\(219\) 0 0
\(220\) −6.51056 −0.438942
\(221\) −2.04223 −0.137375
\(222\) 0 0
\(223\) −15.7307 −1.05341 −0.526704 0.850049i \(-0.676572\pi\)
−0.526704 + 0.850049i \(0.676572\pi\)
\(224\) −2.91729 −0.194919
\(225\) 0 0
\(226\) −9.15859 −0.609220
\(227\) −12.8836 −0.855117 −0.427559 0.903988i \(-0.640626\pi\)
−0.427559 + 0.903988i \(0.640626\pi\)
\(228\) 0 0
\(229\) 5.25383 0.347183 0.173591 0.984818i \(-0.444463\pi\)
0.173591 + 0.984818i \(0.444463\pi\)
\(230\) 5.02112 0.331083
\(231\) 0 0
\(232\) −5.32401 −0.349539
\(233\) −28.3172 −1.85512 −0.927560 0.373675i \(-0.878098\pi\)
−0.927560 + 0.373675i \(0.878098\pi\)
\(234\) 0 0
\(235\) 5.42784 0.354073
\(236\) 1.42784 0.0929447
\(237\) 0 0
\(238\) −7.32401 −0.474745
\(239\) −14.3788 −0.930085 −0.465043 0.885288i \(-0.653961\pi\)
−0.465043 + 0.885288i \(0.653961\pi\)
\(240\) 0 0
\(241\) 18.2749 1.17719 0.588596 0.808427i \(-0.299681\pi\)
0.588596 + 0.808427i \(0.299681\pi\)
\(242\) −31.3874 −2.01766
\(243\) 0 0
\(244\) −1.32401 −0.0847613
\(245\) 1.51056 0.0965060
\(246\) 0 0
\(247\) −0.330856 −0.0210519
\(248\) 8.75186 0.555744
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 1.22019 0.0770174 0.0385087 0.999258i \(-0.487739\pi\)
0.0385087 + 0.999258i \(0.487739\pi\)
\(252\) 0 0
\(253\) 32.6903 2.05522
\(254\) 8.61439 0.540515
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.1306 −1.69236 −0.846181 0.532895i \(-0.821104\pi\)
−0.846181 + 0.532895i \(0.821104\pi\)
\(258\) 0 0
\(259\) −2.91729 −0.181271
\(260\) −0.813457 −0.0504485
\(261\) 0 0
\(262\) 13.4278 0.829575
\(263\) 20.2133 1.24641 0.623204 0.782059i \(-0.285831\pi\)
0.623204 + 0.782059i \(0.285831\pi\)
\(264\) 0 0
\(265\) 2.34513 0.144060
\(266\) −1.18654 −0.0727516
\(267\) 0 0
\(268\) −5.42784 −0.331558
\(269\) 0.207658 0.0126611 0.00633057 0.999980i \(-0.497985\pi\)
0.00633057 + 0.999980i \(0.497985\pi\)
\(270\) 0 0
\(271\) −30.6480 −1.86174 −0.930868 0.365357i \(-0.880947\pi\)
−0.930868 + 0.365357i \(0.880947\pi\)
\(272\) 2.51056 0.152225
\(273\) 0 0
\(274\) −14.6903 −0.887471
\(275\) −6.51056 −0.392601
\(276\) 0 0
\(277\) −16.8135 −1.01022 −0.505111 0.863054i \(-0.668549\pi\)
−0.505111 + 0.863054i \(0.668549\pi\)
\(278\) −17.3662 −1.04156
\(279\) 0 0
\(280\) −2.91729 −0.174341
\(281\) 14.2749 0.851572 0.425786 0.904824i \(-0.359998\pi\)
0.425786 + 0.904824i \(0.359998\pi\)
\(282\) 0 0
\(283\) −13.2288 −0.786369 −0.393184 0.919460i \(-0.628627\pi\)
−0.393184 + 0.919460i \(0.628627\pi\)
\(284\) −14.6480 −0.869201
\(285\) 0 0
\(286\) −5.29606 −0.313162
\(287\) −18.5106 −1.09264
\(288\) 0 0
\(289\) −10.6971 −0.629241
\(290\) −5.32401 −0.312637
\(291\) 0 0
\(292\) −11.0211 −0.644962
\(293\) −2.34513 −0.137004 −0.0685020 0.997651i \(-0.521822\pi\)
−0.0685020 + 0.997651i \(0.521822\pi\)
\(294\) 0 0
\(295\) 1.42784 0.0831323
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −0.207658 −0.0120293
\(299\) 4.08446 0.236210
\(300\) 0 0
\(301\) −21.3662 −1.23153
\(302\) −0.813457 −0.0468092
\(303\) 0 0
\(304\) 0.406728 0.0233275
\(305\) −1.32401 −0.0758128
\(306\) 0 0
\(307\) −23.2624 −1.32766 −0.663828 0.747885i \(-0.731069\pi\)
−0.663828 + 0.747885i \(0.731069\pi\)
\(308\) −18.9932 −1.08224
\(309\) 0 0
\(310\) 8.75186 0.497072
\(311\) 31.3999 1.78052 0.890262 0.455449i \(-0.150521\pi\)
0.890262 + 0.455449i \(0.150521\pi\)
\(312\) 0 0
\(313\) −8.04223 −0.454574 −0.227287 0.973828i \(-0.572986\pi\)
−0.227287 + 0.973828i \(0.572986\pi\)
\(314\) −5.32401 −0.300452
\(315\) 0 0
\(316\) −1.75870 −0.0989346
\(317\) 12.3029 0.691000 0.345500 0.938419i \(-0.387709\pi\)
0.345500 + 0.938419i \(0.387709\pi\)
\(318\) 0 0
\(319\) −34.6623 −1.94072
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 14.6480 0.816303
\(323\) 1.02112 0.0568164
\(324\) 0 0
\(325\) −0.813457 −0.0451225
\(326\) −15.2008 −0.841895
\(327\) 0 0
\(328\) 6.34513 0.350351
\(329\) 15.8346 0.872988
\(330\) 0 0
\(331\) −6.77981 −0.372652 −0.186326 0.982488i \(-0.559658\pi\)
−0.186326 + 0.982488i \(0.559658\pi\)
\(332\) 7.05476 0.387180
\(333\) 0 0
\(334\) −12.4826 −0.683018
\(335\) −5.42784 −0.296555
\(336\) 0 0
\(337\) 10.6903 0.582336 0.291168 0.956672i \(-0.405956\pi\)
0.291168 + 0.956672i \(0.405956\pi\)
\(338\) 12.3383 0.671114
\(339\) 0 0
\(340\) 2.51056 0.136154
\(341\) 56.9795 3.08561
\(342\) 0 0
\(343\) −16.0143 −0.864689
\(344\) 7.32401 0.394884
\(345\) 0 0
\(346\) 23.0354 1.23839
\(347\) −5.62691 −0.302069 −0.151034 0.988529i \(-0.548260\pi\)
−0.151034 + 0.988529i \(0.548260\pi\)
\(348\) 0 0
\(349\) 15.1306 0.809924 0.404962 0.914334i \(-0.367285\pi\)
0.404962 + 0.914334i \(0.367285\pi\)
\(350\) −2.91729 −0.155935
\(351\) 0 0
\(352\) 6.51056 0.347014
\(353\) −35.7816 −1.90446 −0.952230 0.305381i \(-0.901216\pi\)
−0.952230 + 0.305381i \(0.901216\pi\)
\(354\) 0 0
\(355\) −14.6480 −0.777437
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 24.2835 1.28342
\(359\) −4.48260 −0.236583 −0.118291 0.992979i \(-0.537742\pi\)
−0.118291 + 0.992979i \(0.537742\pi\)
\(360\) 0 0
\(361\) −18.8346 −0.991293
\(362\) −2.16543 −0.113812
\(363\) 0 0
\(364\) −2.37309 −0.124384
\(365\) −11.0211 −0.576872
\(366\) 0 0
\(367\) −21.5653 −1.12570 −0.562850 0.826559i \(-0.690295\pi\)
−0.562850 + 0.826559i \(0.690295\pi\)
\(368\) −5.02112 −0.261744
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 6.84141 0.355188
\(372\) 0 0
\(373\) 16.9230 0.876238 0.438119 0.898917i \(-0.355645\pi\)
0.438119 + 0.898917i \(0.355645\pi\)
\(374\) 16.3451 0.845187
\(375\) 0 0
\(376\) −5.42784 −0.279920
\(377\) −4.33086 −0.223050
\(378\) 0 0
\(379\) 31.2710 1.60628 0.803142 0.595788i \(-0.203160\pi\)
0.803142 + 0.595788i \(0.203160\pi\)
\(380\) 0.406728 0.0208647
\(381\) 0 0
\(382\) −19.3326 −0.989142
\(383\) −1.62691 −0.0831314 −0.0415657 0.999136i \(-0.513235\pi\)
−0.0415657 + 0.999136i \(0.513235\pi\)
\(384\) 0 0
\(385\) −18.9932 −0.967981
\(386\) 20.0422 1.02012
\(387\) 0 0
\(388\) 2.34513 0.119056
\(389\) 31.6412 1.60427 0.802136 0.597141i \(-0.203697\pi\)
0.802136 + 0.597141i \(0.203697\pi\)
\(390\) 0 0
\(391\) −12.6058 −0.637503
\(392\) −1.51056 −0.0762947
\(393\) 0 0
\(394\) 15.6269 0.787273
\(395\) −1.75870 −0.0884898
\(396\) 0 0
\(397\) −39.2961 −1.97221 −0.986106 0.166116i \(-0.946878\pi\)
−0.986106 + 0.166116i \(0.946878\pi\)
\(398\) −20.2835 −1.01672
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 9.66914 0.482854 0.241427 0.970419i \(-0.422385\pi\)
0.241427 + 0.970419i \(0.422385\pi\)
\(402\) 0 0
\(403\) 7.11926 0.354636
\(404\) 8.20766 0.408346
\(405\) 0 0
\(406\) −15.5317 −0.770824
\(407\) 6.51056 0.322716
\(408\) 0 0
\(409\) −26.2749 −1.29921 −0.649606 0.760271i \(-0.725066\pi\)
−0.649606 + 0.760271i \(0.725066\pi\)
\(410\) 6.34513 0.313364
\(411\) 0 0
\(412\) −8.81346 −0.434208
\(413\) 4.16543 0.204967
\(414\) 0 0
\(415\) 7.05476 0.346304
\(416\) 0.813457 0.0398830
\(417\) 0 0
\(418\) 2.64803 0.129519
\(419\) 30.1940 1.47507 0.737536 0.675308i \(-0.235989\pi\)
0.737536 + 0.675308i \(0.235989\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −18.7182 −0.911188
\(423\) 0 0
\(424\) −2.34513 −0.113890
\(425\) 2.51056 0.121780
\(426\) 0 0
\(427\) −3.86253 −0.186921
\(428\) −15.2624 −0.737737
\(429\) 0 0
\(430\) 7.32401 0.353195
\(431\) −25.5653 −1.23144 −0.615719 0.787966i \(-0.711134\pi\)
−0.615719 + 0.787966i \(0.711134\pi\)
\(432\) 0 0
\(433\) 32.0422 1.53985 0.769926 0.638134i \(-0.220293\pi\)
0.769926 + 0.638134i \(0.220293\pi\)
\(434\) 25.5317 1.22556
\(435\) 0 0
\(436\) 4.30290 0.206072
\(437\) −2.04223 −0.0976931
\(438\) 0 0
\(439\) 3.93840 0.187970 0.0939848 0.995574i \(-0.470039\pi\)
0.0939848 + 0.995574i \(0.470039\pi\)
\(440\) 6.51056 0.310379
\(441\) 0 0
\(442\) 2.04223 0.0971390
\(443\) 0.0758724 0.00360481 0.00180240 0.999998i \(-0.499426\pi\)
0.00180240 + 0.999998i \(0.499426\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 15.7307 0.744872
\(447\) 0 0
\(448\) 2.91729 0.137829
\(449\) −19.2961 −0.910637 −0.455319 0.890329i \(-0.650475\pi\)
−0.455319 + 0.890329i \(0.650475\pi\)
\(450\) 0 0
\(451\) 41.3103 1.94523
\(452\) 9.15859 0.430784
\(453\) 0 0
\(454\) 12.8836 0.604659
\(455\) −2.37309 −0.111252
\(456\) 0 0
\(457\) 3.00684 0.140654 0.0703271 0.997524i \(-0.477596\pi\)
0.0703271 + 0.997524i \(0.477596\pi\)
\(458\) −5.25383 −0.245495
\(459\) 0 0
\(460\) −5.02112 −0.234111
\(461\) 0.633755 0.0295169 0.0147585 0.999891i \(-0.495302\pi\)
0.0147585 + 0.999891i \(0.495302\pi\)
\(462\) 0 0
\(463\) 18.0422 0.838494 0.419247 0.907872i \(-0.362294\pi\)
0.419247 + 0.907872i \(0.362294\pi\)
\(464\) 5.32401 0.247161
\(465\) 0 0
\(466\) 28.3172 1.31177
\(467\) 23.4758 1.08633 0.543164 0.839626i \(-0.317226\pi\)
0.543164 + 0.839626i \(0.317226\pi\)
\(468\) 0 0
\(469\) −15.8346 −0.731173
\(470\) −5.42784 −0.250368
\(471\) 0 0
\(472\) −1.42784 −0.0657218
\(473\) 47.6834 2.19249
\(474\) 0 0
\(475\) 0.406728 0.0186620
\(476\) 7.32401 0.335696
\(477\) 0 0
\(478\) 14.3788 0.657670
\(479\) 8.19907 0.374625 0.187313 0.982300i \(-0.440022\pi\)
0.187313 + 0.982300i \(0.440022\pi\)
\(480\) 0 0
\(481\) 0.813457 0.0370904
\(482\) −18.2749 −0.832401
\(483\) 0 0
\(484\) 31.3874 1.42670
\(485\) 2.34513 0.106487
\(486\) 0 0
\(487\) 0.953831 0.0432222 0.0216111 0.999766i \(-0.493120\pi\)
0.0216111 + 0.999766i \(0.493120\pi\)
\(488\) 1.32401 0.0599353
\(489\) 0 0
\(490\) −1.51056 −0.0682400
\(491\) −19.3383 −0.872725 −0.436362 0.899771i \(-0.643733\pi\)
−0.436362 + 0.899771i \(0.643733\pi\)
\(492\) 0 0
\(493\) 13.3662 0.601985
\(494\) 0.330856 0.0148859
\(495\) 0 0
\(496\) −8.75186 −0.392970
\(497\) −42.7325 −1.91681
\(498\) 0 0
\(499\) −5.63550 −0.252280 −0.126140 0.992012i \(-0.540259\pi\)
−0.126140 + 0.992012i \(0.540259\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −1.22019 −0.0544595
\(503\) 33.6269 1.49935 0.749675 0.661806i \(-0.230210\pi\)
0.749675 + 0.661806i \(0.230210\pi\)
\(504\) 0 0
\(505\) 8.20766 0.365236
\(506\) −32.6903 −1.45326
\(507\) 0 0
\(508\) −8.61439 −0.382202
\(509\) 18.6903 0.828431 0.414216 0.910179i \(-0.364056\pi\)
0.414216 + 0.910179i \(0.364056\pi\)
\(510\) 0 0
\(511\) −32.1517 −1.42231
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 27.1306 1.19668
\(515\) −8.81346 −0.388367
\(516\) 0 0
\(517\) −35.3383 −1.55418
\(518\) 2.91729 0.128178
\(519\) 0 0
\(520\) 0.813457 0.0356724
\(521\) −25.1586 −1.10222 −0.551109 0.834433i \(-0.685795\pi\)
−0.551109 + 0.834433i \(0.685795\pi\)
\(522\) 0 0
\(523\) −7.25383 −0.317188 −0.158594 0.987344i \(-0.550696\pi\)
−0.158594 + 0.987344i \(0.550696\pi\)
\(524\) −13.4278 −0.586598
\(525\) 0 0
\(526\) −20.2133 −0.881344
\(527\) −21.9720 −0.957117
\(528\) 0 0
\(529\) 2.21160 0.0961564
\(530\) −2.34513 −0.101866
\(531\) 0 0
\(532\) 1.18654 0.0514432
\(533\) 5.16149 0.223569
\(534\) 0 0
\(535\) −15.2624 −0.659852
\(536\) 5.42784 0.234447
\(537\) 0 0
\(538\) −0.207658 −0.00895278
\(539\) −9.83457 −0.423605
\(540\) 0 0
\(541\) 16.0422 0.689709 0.344855 0.938656i \(-0.387928\pi\)
0.344855 + 0.938656i \(0.387928\pi\)
\(542\) 30.6480 1.31645
\(543\) 0 0
\(544\) −2.51056 −0.107639
\(545\) 4.30290 0.184316
\(546\) 0 0
\(547\) 16.0143 0.684721 0.342360 0.939569i \(-0.388774\pi\)
0.342360 + 0.939569i \(0.388774\pi\)
\(548\) 14.6903 0.627537
\(549\) 0 0
\(550\) 6.51056 0.277611
\(551\) 2.16543 0.0922503
\(552\) 0 0
\(553\) −5.13063 −0.218177
\(554\) 16.8135 0.714335
\(555\) 0 0
\(556\) 17.3662 0.736493
\(557\) 10.8557 0.459970 0.229985 0.973194i \(-0.426132\pi\)
0.229985 + 0.973194i \(0.426132\pi\)
\(558\) 0 0
\(559\) 5.95777 0.251987
\(560\) 2.91729 0.123278
\(561\) 0 0
\(562\) −14.2749 −0.602152
\(563\) −31.0776 −1.30977 −0.654883 0.755731i \(-0.727282\pi\)
−0.654883 + 0.755731i \(0.727282\pi\)
\(564\) 0 0
\(565\) 9.15859 0.385305
\(566\) 13.2288 0.556047
\(567\) 0 0
\(568\) 14.6480 0.614618
\(569\) −30.0845 −1.26121 −0.630603 0.776105i \(-0.717192\pi\)
−0.630603 + 0.776105i \(0.717192\pi\)
\(570\) 0 0
\(571\) 5.55673 0.232542 0.116271 0.993218i \(-0.462906\pi\)
0.116271 + 0.993218i \(0.462906\pi\)
\(572\) 5.29606 0.221439
\(573\) 0 0
\(574\) 18.5106 0.772616
\(575\) −5.02112 −0.209395
\(576\) 0 0
\(577\) −22.2499 −0.926275 −0.463137 0.886286i \(-0.653276\pi\)
−0.463137 + 0.886286i \(0.653276\pi\)
\(578\) 10.6971 0.444941
\(579\) 0 0
\(580\) 5.32401 0.221068
\(581\) 20.5807 0.853833
\(582\) 0 0
\(583\) −15.2681 −0.632340
\(584\) 11.0211 0.456057
\(585\) 0 0
\(586\) 2.34513 0.0968764
\(587\) −45.3103 −1.87016 −0.935079 0.354440i \(-0.884672\pi\)
−0.935079 + 0.354440i \(0.884672\pi\)
\(588\) 0 0
\(589\) −3.55963 −0.146672
\(590\) −1.42784 −0.0587834
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 0.232712 0.00955635 0.00477817 0.999989i \(-0.498479\pi\)
0.00477817 + 0.999989i \(0.498479\pi\)
\(594\) 0 0
\(595\) 7.32401 0.300255
\(596\) 0.207658 0.00850601
\(597\) 0 0
\(598\) −4.08446 −0.167026
\(599\) −43.4056 −1.77350 −0.886752 0.462246i \(-0.847044\pi\)
−0.886752 + 0.462246i \(0.847044\pi\)
\(600\) 0 0
\(601\) 35.5317 1.44937 0.724684 0.689082i \(-0.241986\pi\)
0.724684 + 0.689082i \(0.241986\pi\)
\(602\) 21.3662 0.870823
\(603\) 0 0
\(604\) 0.813457 0.0330991
\(605\) 31.3874 1.27608
\(606\) 0 0
\(607\) 26.7325 1.08504 0.542519 0.840043i \(-0.317471\pi\)
0.542519 + 0.840043i \(0.317471\pi\)
\(608\) −0.406728 −0.0164950
\(609\) 0 0
\(610\) 1.32401 0.0536078
\(611\) −4.41532 −0.178625
\(612\) 0 0
\(613\) 48.0565 1.94098 0.970492 0.241134i \(-0.0775192\pi\)
0.970492 + 0.241134i \(0.0775192\pi\)
\(614\) 23.2624 0.938795
\(615\) 0 0
\(616\) 18.9932 0.765256
\(617\) −12.9789 −0.522510 −0.261255 0.965270i \(-0.584136\pi\)
−0.261255 + 0.965270i \(0.584136\pi\)
\(618\) 0 0
\(619\) 25.3662 1.01956 0.509778 0.860306i \(-0.329728\pi\)
0.509778 + 0.860306i \(0.329728\pi\)
\(620\) −8.75186 −0.351483
\(621\) 0 0
\(622\) −31.3999 −1.25902
\(623\) −17.5037 −0.701272
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.04223 0.321432
\(627\) 0 0
\(628\) 5.32401 0.212451
\(629\) −2.51056 −0.100102
\(630\) 0 0
\(631\) −31.6075 −1.25828 −0.629138 0.777293i \(-0.716592\pi\)
−0.629138 + 0.777293i \(0.716592\pi\)
\(632\) 1.75870 0.0699573
\(633\) 0 0
\(634\) −12.3029 −0.488611
\(635\) −8.61439 −0.341852
\(636\) 0 0
\(637\) −1.22877 −0.0486858
\(638\) 34.6623 1.37229
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 23.8625 0.942513 0.471257 0.881996i \(-0.343801\pi\)
0.471257 + 0.881996i \(0.343801\pi\)
\(642\) 0 0
\(643\) 32.8277 1.29460 0.647300 0.762236i \(-0.275898\pi\)
0.647300 + 0.762236i \(0.275898\pi\)
\(644\) −14.6480 −0.577213
\(645\) 0 0
\(646\) −1.02112 −0.0401752
\(647\) 13.9578 0.548737 0.274368 0.961625i \(-0.411531\pi\)
0.274368 + 0.961625i \(0.411531\pi\)
\(648\) 0 0
\(649\) −9.29606 −0.364902
\(650\) 0.813457 0.0319064
\(651\) 0 0
\(652\) 15.2008 0.595310
\(653\) −28.5921 −1.11890 −0.559448 0.828865i \(-0.688987\pi\)
−0.559448 + 0.828865i \(0.688987\pi\)
\(654\) 0 0
\(655\) −13.4278 −0.524669
\(656\) −6.34513 −0.247736
\(657\) 0 0
\(658\) −15.8346 −0.617296
\(659\) 5.62691 0.219193 0.109597 0.993976i \(-0.465044\pi\)
0.109597 + 0.993976i \(0.465044\pi\)
\(660\) 0 0
\(661\) 13.3240 0.518244 0.259122 0.965845i \(-0.416567\pi\)
0.259122 + 0.965845i \(0.416567\pi\)
\(662\) 6.77981 0.263505
\(663\) 0 0
\(664\) −7.05476 −0.273778
\(665\) 1.18654 0.0460122
\(666\) 0 0
\(667\) −26.7325 −1.03509
\(668\) 12.4826 0.482966
\(669\) 0 0
\(670\) 5.42784 0.209696
\(671\) 8.62007 0.332774
\(672\) 0 0
\(673\) 29.6691 1.14366 0.571831 0.820372i \(-0.306233\pi\)
0.571831 + 0.820372i \(0.306233\pi\)
\(674\) −10.6903 −0.411773
\(675\) 0 0
\(676\) −12.3383 −0.474550
\(677\) 1.25383 0.0481885 0.0240943 0.999710i \(-0.492330\pi\)
0.0240943 + 0.999710i \(0.492330\pi\)
\(678\) 0 0
\(679\) 6.84141 0.262549
\(680\) −2.51056 −0.0962755
\(681\) 0 0
\(682\) −56.9795 −2.18186
\(683\) 4.76045 0.182153 0.0910767 0.995844i \(-0.470969\pi\)
0.0910767 + 0.995844i \(0.470969\pi\)
\(684\) 0 0
\(685\) 14.6903 0.561286
\(686\) 16.0143 0.611428
\(687\) 0 0
\(688\) −7.32401 −0.279225
\(689\) −1.90766 −0.0726761
\(690\) 0 0
\(691\) −20.2219 −0.769279 −0.384639 0.923067i \(-0.625674\pi\)
−0.384639 + 0.923067i \(0.625674\pi\)
\(692\) −23.0354 −0.875674
\(693\) 0 0
\(694\) 5.62691 0.213595
\(695\) 17.3662 0.658739
\(696\) 0 0
\(697\) −15.9298 −0.603385
\(698\) −15.1306 −0.572703
\(699\) 0 0
\(700\) 2.91729 0.110263
\(701\) 22.0845 0.834119 0.417059 0.908879i \(-0.363061\pi\)
0.417059 + 0.908879i \(0.363061\pi\)
\(702\) 0 0
\(703\) −0.406728 −0.0153401
\(704\) −6.51056 −0.245376
\(705\) 0 0
\(706\) 35.7816 1.34666
\(707\) 23.9441 0.900510
\(708\) 0 0
\(709\) 5.59896 0.210273 0.105137 0.994458i \(-0.466472\pi\)
0.105137 + 0.994458i \(0.466472\pi\)
\(710\) 14.6480 0.549731
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 43.9441 1.64572
\(714\) 0 0
\(715\) 5.29606 0.198061
\(716\) −24.2835 −0.907518
\(717\) 0 0
\(718\) 4.48260 0.167289
\(719\) −40.2921 −1.50264 −0.751321 0.659937i \(-0.770583\pi\)
−0.751321 + 0.659937i \(0.770583\pi\)
\(720\) 0 0
\(721\) −25.7114 −0.957542
\(722\) 18.8346 0.700950
\(723\) 0 0
\(724\) 2.16543 0.0804775
\(725\) 5.32401 0.197729
\(726\) 0 0
\(727\) 11.2538 0.417381 0.208691 0.977982i \(-0.433080\pi\)
0.208691 + 0.977982i \(0.433080\pi\)
\(728\) 2.37309 0.0879524
\(729\) 0 0
\(730\) 11.0211 0.407910
\(731\) −18.3874 −0.680081
\(732\) 0 0
\(733\) −42.2892 −1.56199 −0.780994 0.624539i \(-0.785287\pi\)
−0.780994 + 0.624539i \(0.785287\pi\)
\(734\) 21.5653 0.795990
\(735\) 0 0
\(736\) 5.02112 0.185081
\(737\) 35.3383 1.30170
\(738\) 0 0
\(739\) −29.3103 −1.07820 −0.539099 0.842242i \(-0.681235\pi\)
−0.539099 + 0.842242i \(0.681235\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −6.84141 −0.251156
\(743\) 4.39245 0.161144 0.0805718 0.996749i \(-0.474325\pi\)
0.0805718 + 0.996749i \(0.474325\pi\)
\(744\) 0 0
\(745\) 0.207658 0.00760801
\(746\) −16.9230 −0.619594
\(747\) 0 0
\(748\) −16.3451 −0.597637
\(749\) −44.5248 −1.62690
\(750\) 0 0
\(751\) 2.58074 0.0941727 0.0470864 0.998891i \(-0.485006\pi\)
0.0470864 + 0.998891i \(0.485006\pi\)
\(752\) 5.42784 0.197933
\(753\) 0 0
\(754\) 4.33086 0.157720
\(755\) 0.813457 0.0296047
\(756\) 0 0
\(757\) 27.3805 0.995162 0.497581 0.867418i \(-0.334222\pi\)
0.497581 + 0.867418i \(0.334222\pi\)
\(758\) −31.2710 −1.13581
\(759\) 0 0
\(760\) −0.406728 −0.0147536
\(761\) 42.9509 1.55697 0.778485 0.627663i \(-0.215989\pi\)
0.778485 + 0.627663i \(0.215989\pi\)
\(762\) 0 0
\(763\) 12.5528 0.454441
\(764\) 19.3326 0.699429
\(765\) 0 0
\(766\) 1.62691 0.0587828
\(767\) −1.16149 −0.0419389
\(768\) 0 0
\(769\) −13.0633 −0.471076 −0.235538 0.971865i \(-0.575685\pi\)
−0.235538 + 0.971865i \(0.575685\pi\)
\(770\) 18.9932 0.684466
\(771\) 0 0
\(772\) −20.0422 −0.721336
\(773\) 15.3662 0.552685 0.276343 0.961059i \(-0.410878\pi\)
0.276343 + 0.961059i \(0.410878\pi\)
\(774\) 0 0
\(775\) −8.75186 −0.314376
\(776\) −2.34513 −0.0841852
\(777\) 0 0
\(778\) −31.6412 −1.13439
\(779\) −2.58074 −0.0924648
\(780\) 0 0
\(781\) 95.3668 3.41249
\(782\) 12.6058 0.450782
\(783\) 0 0
\(784\) 1.51056 0.0539485
\(785\) 5.32401 0.190022
\(786\) 0 0
\(787\) 10.9316 0.389668 0.194834 0.980836i \(-0.437583\pi\)
0.194834 + 0.980836i \(0.437583\pi\)
\(788\) −15.6269 −0.556686
\(789\) 0 0
\(790\) 1.75870 0.0625717
\(791\) 26.7182 0.949990
\(792\) 0 0
\(793\) 1.07703 0.0382464
\(794\) 39.2961 1.39456
\(795\) 0 0
\(796\) 20.2835 0.718931
\(797\) 24.2921 0.860471 0.430235 0.902717i \(-0.358431\pi\)
0.430235 + 0.902717i \(0.358431\pi\)
\(798\) 0 0
\(799\) 13.6269 0.482086
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −9.66914 −0.341429
\(803\) 71.7536 2.53213
\(804\) 0 0
\(805\) −14.6480 −0.516275
\(806\) −7.11926 −0.250765
\(807\) 0 0
\(808\) −8.20766 −0.288744
\(809\) 40.0422 1.40781 0.703905 0.710294i \(-0.251438\pi\)
0.703905 + 0.710294i \(0.251438\pi\)
\(810\) 0 0
\(811\) −40.6343 −1.42686 −0.713432 0.700724i \(-0.752860\pi\)
−0.713432 + 0.700724i \(0.752860\pi\)
\(812\) 15.5317 0.545055
\(813\) 0 0
\(814\) −6.51056 −0.228195
\(815\) 15.2008 0.532461
\(816\) 0 0
\(817\) −2.97888 −0.104218
\(818\) 26.2749 0.918682
\(819\) 0 0
\(820\) −6.34513 −0.221582
\(821\) 41.4787 1.44762 0.723808 0.690002i \(-0.242390\pi\)
0.723808 + 0.690002i \(0.242390\pi\)
\(822\) 0 0
\(823\) −5.27610 −0.183913 −0.0919566 0.995763i \(-0.529312\pi\)
−0.0919566 + 0.995763i \(0.529312\pi\)
\(824\) 8.81346 0.307031
\(825\) 0 0
\(826\) −4.16543 −0.144934
\(827\) −1.76438 −0.0613537 −0.0306768 0.999529i \(-0.509766\pi\)
−0.0306768 + 0.999529i \(0.509766\pi\)
\(828\) 0 0
\(829\) 16.3029 0.566223 0.283112 0.959087i \(-0.408633\pi\)
0.283112 + 0.959087i \(0.408633\pi\)
\(830\) −7.05476 −0.244874
\(831\) 0 0
\(832\) −0.813457 −0.0282015
\(833\) 3.79234 0.131397
\(834\) 0 0
\(835\) 12.4826 0.431978
\(836\) −2.64803 −0.0915840
\(837\) 0 0
\(838\) −30.1940 −1.04303
\(839\) −14.3731 −0.496214 −0.248107 0.968733i \(-0.579808\pi\)
−0.248107 + 0.968733i \(0.579808\pi\)
\(840\) 0 0
\(841\) −0.654870 −0.0225817
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 18.7182 0.644308
\(845\) −12.3383 −0.424450
\(846\) 0 0
\(847\) 91.5659 3.14624
\(848\) 2.34513 0.0805321
\(849\) 0 0
\(850\) −2.51056 −0.0861114
\(851\) 5.02112 0.172122
\(852\) 0 0
\(853\) 43.2961 1.48243 0.741214 0.671268i \(-0.234250\pi\)
0.741214 + 0.671268i \(0.234250\pi\)
\(854\) 3.86253 0.132173
\(855\) 0 0
\(856\) 15.2624 0.521659
\(857\) 49.0776 1.67646 0.838230 0.545317i \(-0.183591\pi\)
0.838230 + 0.545317i \(0.183591\pi\)
\(858\) 0 0
\(859\) 15.5374 0.530128 0.265064 0.964231i \(-0.414607\pi\)
0.265064 + 0.964231i \(0.414607\pi\)
\(860\) −7.32401 −0.249747
\(861\) 0 0
\(862\) 25.5653 0.870758
\(863\) 36.0901 1.22852 0.614261 0.789103i \(-0.289454\pi\)
0.614261 + 0.789103i \(0.289454\pi\)
\(864\) 0 0
\(865\) −23.0354 −0.783227
\(866\) −32.0422 −1.08884
\(867\) 0 0
\(868\) −25.5317 −0.866601
\(869\) 11.4501 0.388419
\(870\) 0 0
\(871\) 4.41532 0.149607
\(872\) −4.30290 −0.145715
\(873\) 0 0
\(874\) 2.04223 0.0690795
\(875\) 2.91729 0.0986223
\(876\) 0 0
\(877\) 29.3240 0.990202 0.495101 0.868836i \(-0.335131\pi\)
0.495101 + 0.868836i \(0.335131\pi\)
\(878\) −3.93840 −0.132915
\(879\) 0 0
\(880\) −6.51056 −0.219471
\(881\) −23.8066 −0.802065 −0.401033 0.916064i \(-0.631349\pi\)
−0.401033 + 0.916064i \(0.631349\pi\)
\(882\) 0 0
\(883\) −3.94699 −0.132827 −0.0664134 0.997792i \(-0.521156\pi\)
−0.0664134 + 0.997792i \(0.521156\pi\)
\(884\) −2.04223 −0.0686876
\(885\) 0 0
\(886\) −0.0758724 −0.00254898
\(887\) −1.96346 −0.0659264 −0.0329632 0.999457i \(-0.510494\pi\)
−0.0329632 + 0.999457i \(0.510494\pi\)
\(888\) 0 0
\(889\) −25.1306 −0.842854
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −15.7307 −0.526704
\(893\) 2.20766 0.0738765
\(894\) 0 0
\(895\) −24.2835 −0.811709
\(896\) −2.91729 −0.0974597
\(897\) 0 0
\(898\) 19.2961 0.643918
\(899\) −46.5950 −1.55403
\(900\) 0 0
\(901\) 5.88758 0.196144
\(902\) −41.3103 −1.37548
\(903\) 0 0
\(904\) −9.15859 −0.304610
\(905\) 2.16543 0.0719813
\(906\) 0 0
\(907\) 5.89049 0.195590 0.0977952 0.995207i \(-0.468821\pi\)
0.0977952 + 0.995207i \(0.468821\pi\)
\(908\) −12.8836 −0.427559
\(909\) 0 0
\(910\) 2.37309 0.0786670
\(911\) 27.9527 0.926113 0.463057 0.886329i \(-0.346753\pi\)
0.463057 + 0.886329i \(0.346753\pi\)
\(912\) 0 0
\(913\) −45.9304 −1.52007
\(914\) −3.00684 −0.0994575
\(915\) 0 0
\(916\) 5.25383 0.173591
\(917\) −39.1729 −1.29360
\(918\) 0 0
\(919\) 17.7587 0.585805 0.292903 0.956142i \(-0.405379\pi\)
0.292903 + 0.956142i \(0.405379\pi\)
\(920\) 5.02112 0.165541
\(921\) 0 0
\(922\) −0.633755 −0.0208716
\(923\) 11.9155 0.392205
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −18.0422 −0.592904
\(927\) 0 0
\(928\) −5.32401 −0.174769
\(929\) −34.2892 −1.12499 −0.562496 0.826800i \(-0.690159\pi\)
−0.562496 + 0.826800i \(0.690159\pi\)
\(930\) 0 0
\(931\) 0.614387 0.0201357
\(932\) −28.3172 −0.927560
\(933\) 0 0
\(934\) −23.4758 −0.768150
\(935\) −16.3451 −0.534543
\(936\) 0 0
\(937\) −14.0845 −0.460119 −0.230060 0.973177i \(-0.573892\pi\)
−0.230060 + 0.973177i \(0.573892\pi\)
\(938\) 15.8346 0.517017
\(939\) 0 0
\(940\) 5.42784 0.177037
\(941\) −14.2499 −0.464533 −0.232267 0.972652i \(-0.574614\pi\)
−0.232267 + 0.972652i \(0.574614\pi\)
\(942\) 0 0
\(943\) 31.8596 1.03749
\(944\) 1.42784 0.0464723
\(945\) 0 0
\(946\) −47.6834 −1.55032
\(947\) 29.8203 0.969029 0.484515 0.874783i \(-0.338996\pi\)
0.484515 + 0.874783i \(0.338996\pi\)
\(948\) 0 0
\(949\) 8.96520 0.291023
\(950\) −0.406728 −0.0131960
\(951\) 0 0
\(952\) −7.32401 −0.237373
\(953\) −8.64803 −0.280137 −0.140069 0.990142i \(-0.544732\pi\)
−0.140069 + 0.990142i \(0.544732\pi\)
\(954\) 0 0
\(955\) 19.3326 0.625588
\(956\) −14.3788 −0.465043
\(957\) 0 0
\(958\) −8.19907 −0.264900
\(959\) 42.8557 1.38388
\(960\) 0 0
\(961\) 45.5950 1.47081
\(962\) −0.813457 −0.0262269
\(963\) 0 0
\(964\) 18.2749 0.588596
\(965\) −20.0422 −0.645182
\(966\) 0 0
\(967\) −45.0325 −1.44815 −0.724074 0.689723i \(-0.757732\pi\)
−0.724074 + 0.689723i \(0.757732\pi\)
\(968\) −31.3874 −1.00883
\(969\) 0 0
\(970\) −2.34513 −0.0752976
\(971\) −26.3029 −0.844100 −0.422050 0.906573i \(-0.638689\pi\)
−0.422050 + 0.906573i \(0.638689\pi\)
\(972\) 0 0
\(973\) 50.6623 1.62416
\(974\) −0.953831 −0.0305627
\(975\) 0 0
\(976\) −1.32401 −0.0423807
\(977\) −25.6549 −0.820772 −0.410386 0.911912i \(-0.634606\pi\)
−0.410386 + 0.911912i \(0.634606\pi\)
\(978\) 0 0
\(979\) 39.0633 1.24847
\(980\) 1.51056 0.0482530
\(981\) 0 0
\(982\) 19.3383 0.617110
\(983\) 55.3611 1.76575 0.882873 0.469611i \(-0.155606\pi\)
0.882873 + 0.469611i \(0.155606\pi\)
\(984\) 0 0
\(985\) −15.6269 −0.497915
\(986\) −13.3662 −0.425668
\(987\) 0 0
\(988\) −0.330856 −0.0105259
\(989\) 36.7747 1.16937
\(990\) 0 0
\(991\) −3.24814 −0.103181 −0.0515903 0.998668i \(-0.516429\pi\)
−0.0515903 + 0.998668i \(0.516429\pi\)
\(992\) 8.75186 0.277872
\(993\) 0 0
\(994\) 42.7325 1.35539
\(995\) 20.2835 0.643031
\(996\) 0 0
\(997\) −12.6480 −0.400567 −0.200284 0.979738i \(-0.564186\pi\)
−0.200284 + 0.979738i \(0.564186\pi\)
\(998\) 5.63550 0.178389
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3330.2.a.bg.1.3 3
3.2 odd 2 370.2.a.g.1.2 3
12.11 even 2 2960.2.a.u.1.2 3
15.2 even 4 1850.2.b.o.149.5 6
15.8 even 4 1850.2.b.o.149.2 6
15.14 odd 2 1850.2.a.z.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.2 3 3.2 odd 2
1850.2.a.z.1.2 3 15.14 odd 2
1850.2.b.o.149.2 6 15.8 even 4
1850.2.b.o.149.5 6 15.2 even 4
2960.2.a.u.1.2 3 12.11 even 2
3330.2.a.bg.1.3 3 1.1 even 1 trivial